Some Properties Of N-Dimensional M(?)bius Groups

The main purpose of this thesis is to investigate properties of n-dimensional Mobius groups. This thesis is arranged as follows.Chapter 1 provides some background information about Mobius groups and statement of our main results.In Chapter 2, we introduce some basic concepts and properties of Mobius groups, including the Clifford algebra, Clifford matrices , classification of elements in M(Rn) etc.In Chapter 3, by using the traces of the generators / and g of two-generator Mobius subgroup (f, g), we define three-number sets (β(f),β(g),r(f, g)) in C× C× C and investigate the relationship among discreteness, elementariness and three-number sets. Some properties of the set E in C x C x C are given, and a precise description of the set D E1 and some properties of the three-number sets corresponding to two-generator elementary discrete Mobius subgroups are presented. Moreover, by using algebraic method, we prove that the sets E and D E are closed in C x C x C.Beardon A.F. etc discussed the conjugation between a subgroup in SL(2, C) and one in SL(2, R) and obtained a sufficient and necessary condition and some other sufficient conditions. In Chapter 4, we study this problem further and prove five necessary and sufficient conditions for a subgroup in SL(2, C) to be conjugate to one in SL(2,R). These generalize the known corresponding results. Also we generalize these discussions into the higer dimensional cases and one necessary and sufficient condition for a group in SL(2,Tn} to be conjugate to one in SL(2,R). Our result generalizes those obtained by Maskit B. etc.In Chapter 5, after by weakening the assumptions of the inequalities about high dimensional Mobius subgroups containing strictly parabolic element obtained by Hersonsky S., Leutbecher A., Ohtake H., Shimizu H. etc, we establish a new one. Finally, two applications of this inequality are given.